A q-Raabe formula and an integral of the fourth Jacobi theta function

Aq-Raabe formula and an integral of the fourth Jacobi theta function Istv´an Mez˝o Escuela Polit´ecnica Nacional, Departamento de Matem´atica, Ladr´on de Guevara E11-253, Quito, Ecuador Abstract We generalize the Raabe-formula to the q-loggamma function by giving an integral formula for log Γqwhen q > 1. As a consequence, we get that the integral of the logarithm of the fourth Jacobi theta function between its least imaginary zeros is connected to the partition function and the Riemann zeta function. Key words: q-gamma function, q-loggamma function, Jacobi theta functions, hypergeometric function, Riemann zeta function, partition function, Raabe-formula 1991 MSC: 33E05, 33D05 1 Introduction The fourth Jacobi theta function is defined by the infinite sums ϑ4(x, q) = ∞ X n=−∞ (−1)nqn2e2nix = 1 + 2 ∞ X n=1 (−1)nqn2cos(2nx). This is an entire function in the complex variable xfor any fixed complex q for which |q|<1. See [4,25] on the theta functions in general. The partition function P(n) gives the number of possible additive integer partitions of the natural number n. In other words, P(n) is the number of ways writing nas a sum of positive integers [3,11]. Email address: [email protected] (Istv´an Mez˝o). Preprint submitted to Journal of Number Theory 29 August 2012 In the present paper we reveal an analytic connection between the fourth Jacobi function and the partition function. It is easy to prove that for a fixed real q∈]0,1[ the function ϑ4(ix, q) is real and strictly positive when x∈] log √q, −log √q[, and it is zero at these two endpoints (see the final remarks of Section 4). Our main result concerns on the area under the graph of the fourth Jacobi function on this interval (the graph can be found in the penultimate section). Theorem 1 For any real q∈]0,1[, Zx∗ −x∗ log ϑ4(ix, q)dx =ζ(2) −log q·log ∞ X n=0 P(n)q2n!. Here x∗= log √q,P(n)is the partition function, and i=√−1. (Note that x∗is negative and −x∗is positive, so in the theorem the integration goes from ”right to left” on the imaginary axis.) To prove the theorem we need a q-analogue of Raabe’s integral Z1 0log Γ(x)dx = log √2π for the Euler gamma function. This analogue uses the so-called q-gamma function and states: Z1 0log Γq(x)dx =ζ(2) log q+ log v u u t q−1 6 √q+ log(q−1;q−1)∞(q > 1), where (q−1;q−1)∞=Q∞ n=0(1 −1/qn). For a more general statement, see Theorem 2. The definition of Γqis given in equation (2). 2 Preliminaries In this section we set up all the necessary ingredients and preliminary propositions to prove our q-Raabe formula and the integral formula. 2 2.1 Raabe’s formula In 1840 J. L. Raabe [22] proved that for the Euler Γ function Z1 0log Γ(x+t)dx = log √2π+tlog t−t(t≥0). This implies the special case (when we take the limit t→0+) Z1 0log Γ(x)dx = log √2π, and an immediate consequence is that Z1 0log Γ(x)Γ(1 −x)dx = log 2π. (See [1] for an elementary proof of this special case.) We shall prove the appropriate integral formula for the q-gamma function when q > 1 (Theorem 2.). Then we show that the Jacobi triple product identity connects the q-gamma function to ϑ4and our main formula (Theorem 1.) will follow. To read more on the Raabe-formula and its extension to multidimensional case, the reader may consult [16,23]. 2.2 The q-gamma functions F. H. Jackson defined, for 0 < q < 1, the q-analogue of the standard Euler Γ(x) function for any x∈R\{0,−1,−2, . . . }as [8,13,12] Γq(x) = (q;q)∞ (qx;q)∞ (1 −q)1−x(0 < q < 1) (1) with the so-called q-Pochhammer symbol (x;q)∞= (1−x)(1−qx)(1−q2x)···. This Γqfunction is called as Jackson q-gamma function. This plays an important role in the evaluation of basic hypergeometric series [8]. R. Askey also contributed to the Jackson q-gamma function in a profound way, see [5,6] for examples. On analytic properties of Γq(including information on poles, residues, infinite sum representations) one can turn to the book [24, Section 6.4]. Another q-gamma function can also be defined for q > 1. It is Γq(x) = (q−1;q−1)∞ (q−x;q−1)∞ (q−1)1−xq(x 2)(q > 1).(2) 3 This function was introduced by Jackson [12, p. 129], but he did not study its properties. There are two fundamental papers of D. S. Moak [20,21], in which he investigated its analytic properties (see also Exercise 1.23 of [8] and Exercise 14 on p. 546 in [24]). Therefore in considering this contribution of Moak, one might call the function in (2) the Moak q-gamma function.5 We emphasize that in the present paper we need and use only the q-gamma function when q > 1. 2.3 The zeta regularized product We also need some recent results of Kurokawa and Wakayama. Let us consider a sequence a= (a1, a2, . . . ). Its zeta regularized product is denoted and defined by [14,15] [ ∞ Y n=1 an= exp −Res s=0 ζa(s) s2!.(3) Here ζa(s) = ∞ X n=1 a−s n is the zeta function associated with the sequence a. It is assumed that ζa(s) is meromorphic at s= 0 or at least it can be meromorphically continued to s= 0, and further, that around this point ζa(s) has the Laurent expansion ζa(s) = X m>m0 cm(a)sm for some integer m0. Thus the zeta regularized product equals to exp(−c1(a)), as well. See [19] for nice applications of regularized products. M. Lerch’s formula (5) implies that for a= (x, 1 + x, 2 + x, . . . ) [ ∞ Y n=0 (n+x) = √2π Γ(x)(x > 0).(4) The sequence aabove has the associated zeta function ζa(s) = ζ(s, x) = ∞ X n=0 (n+x)−s(x > 0,<(s)>1). This is the well known Hurwitz zeta function. What Lerch proved is that [17] ζ0(0, x) = log Γ(x) √2π(x > 0).(5) 4 Then (4) easily follows. See [2, Theorem 1.3.4] for a proof of (5). Now we step forward to the q-version of the above theorems. Let us introduce the short and standard notation [n]q=qn−1 q−1(q6= 1). With this abbreviation the zeta function associated with the sequence a= ([x],[1 + x]q,[2 + x]q, . . . ) is the so-called q-Hurwitz zeta function: ζa(s) = ζq(s, x) = ∞ X n=0 [n+x]−s q. There are different q-extensions of the ordinary Hurwitz zeta function, see [24] for different examples. Since the analytical properties and convergence domains of this function are crucial in the present investigation, we formulate the next proposition. Proposition 1 For any fixed q > 1the q-Hurwitz zeta function ζq(s, x) = ∞ X n=0 [n+x]−s q converges when x > 0and <(s)>0. At the point s= 0 this function has a simple pole with residue 1/log(q). The proof of this proposition can be found in the last section. With respect to the q-gamma function, the parallel result of (4) is in the next proposition. Proposition 2 For any real q > 1and x > 0, there holds [ ∞ Y n=0 [n+x]q= [ ∞ Y n=1 qn+x−1 q−1=Cq Γq(x),(6) where Cq=q−1 12 (q−1)1 2−log(q−1) 2 log q(q−1;q−1)∞.(7) This is the second theorem of Kurokawa and Wakayama in [14] and this will be our main tool. (A more general form of this theorem is presented in [19].) For practical reasons we rephrase this zeta regularization theorem in a more suitable form (employing (3) and (6)). 5 Proposition 3 For any q > 1and x > 0, there holds log Γq(x) = log Cq+ Res s=0 ζq(s, x) s2, where Cqis defined by (7). The statement follows from Proposition 2 by using the regularization formula (3). We split the proof of the main theorem to two sections. The next one contains the generalized Raabe’s formula, the other contains the proof of the integral formula of Jacobi’s function ϑ4. 3 Integral of the q-loggamma function – the q-Raabe formula The q-analogue of Raabe’s theorem for q > 1 is given: Theorem 2 If q > 1and Γq(x)is defined by (2), then for any t > 0, Z1 0log Γq(x+t)dx = (8) log Cq−1 2qtlog q"1−qt 1−q−t(2 Li2(q−t) + log2(1 −q−t))+ 21−qt 1−q−tlog 1−q 1−qtlog(1 −q−t)−qtlog21−q 1−qt#. In particular, if ttends to zero then Z1 0log Γq(x)dx =ζ(2) log q+ log v u u t q−1 6 √q+ log(q−1;q−1)∞.(9) Here Li2(z) is the dilogarithm function [18]: Li2(z) = ∞ X n=1 zn n2. It is an interesting question that how such a theorem looks like when we use the Jackson q-gamma function (i.e., definition (1) and 0 < q < 1). To look for a theorem of this flavour, our proof cannot be applied, since the two crucial points – the q-Hurwitz zeta and the Kurokawa-Wakayama theorem – work only when q > 1. 6 To prove Theorem 2, we need the following statement on the integral of the q-Hurwitz zeta function, which is interesting in itself. Theorem 3 If q > 1,t > 0and <(s)>0, then Z1 0ζq(s, x +t)dx =(q−1)s slog q (qt−1)1−s qt2F1(1,1; s+ 1; q−t). Here 2F1(a, b;c;z) = ∞ X n=0 (a)n(b)n (c)n zn n! is a hypergeometric function and (a)n=a(a+1) ···(a+n−1) is the Pochhammer symbol. See a good introduction to hypergeometric functions in [10]. Proof of Theorem 3. To use the series representation of the q-Hurwitz zeta function, we have to assume that t > 0, q > 1 and <(s)>0 (see Proposition 1). Then we have that Z1 0ζq(s, x +t)dx =Z1 0 ∞ X n=0 [n+x+t]−s qdx = (q−1)s∞ X n=0 Z1 0(qn+x+t−1)−sdx. This latter integral can be computed if we determine the Taylor series of the integrand with respect to the variable xand then we integrate term by term. A lengthy computation finally shows that the integral can be expressed by hypergeometric functions: Z1 0(qn+x+t−1)−sdx =1 slog q"(qn+t−1)1−s qn+t2F1(1,1; s+ 1; q−n−t)− (qn+t+1 −1)1−s qn+t+1 2F1(1,1; s+ 1; q−n−t−1)#. (It can be realized that the above subtraction comes from the Newton-Leibniz formula, so one can read out the primitive function and then check this integral by derivation, too.) Since 2F1(1,1; s+ 1; q−n−t) = ∞ X k=0 k! (s+ 1)k 1 (qn+t)k, and because of <(s)>0, ∞ X n=0 Z1 0(qn+x+t−1)−sdx = 7 1 slog q ∞ X n=0 ∞ X k=0 k! (s+ 1)k (qn+t−1)1−s (qn+t)k+1 −(qn+t+1 −1)1−s (qn+t+1)k+1 !. If we interchange the order of the summation – which can be done by absolute convergence –, we see that the sum over nis telescopic, so the only one term which not cancels belongs to n= 0. Thus the above expression simplifies to 1 slog q ∞ X k=0 k! (s+ 1)k (qt−1)1−s (qt)k+1 =(qt−1)1−s sqtlog q ∞ X k=0 k! (s+ 1)k 1 (qt)k. This latter sum is again hypergeometric with parameters (1,1; s+1; q−t), hence we get our Theorem. 2 Proof of Theorem 2. Proposition 3 gives that Z1 0log Γq(x+t)dx = log Cq+Z1 0Res s=0 ζ(s, x +t) s2dx. Since the residue is taken with respect to s, we can carry out it before the integral. Hence, by Theorem 3, Z1 0log Γq(x+t)dx = log Cq+ Res s=0 (q−1)s s3log q (qt−1)1−s qt2F1(1,1; s+ 1; q−t). The residue can be calculated as follows: we leave s3in the denominator, then we look for the coefficient of s2in the Taylor expansion of the remaining function. A lenghty calculation shows that the residue equals to −1 2qtlog q"(1 −qt)∂2 ∂s22F1(1,1; s;q−t)s=1 + (10) 2(1 −qt) log 1−q 1−qt ∂ ∂s2F1(1,1; s;q−t)s=1 −qtlog21−q 1−qt#. Now we deal with the partial derivatives. Symbolically, ∂n ∂sn2F1(1,1; s;z) = ∞ X n=0 (1)n(1)n ∂n ∂sn 1 (s)n zn n!.(11) The Pochhammer symbol can be rewritten with the Γ function: (s)n=Γ(s+n) Γ(s), whence ∂ ∂s 1 (s)n =−1 (s)n (ψ(s+n)−ψ(s)),(12) and ∂2 ∂s2 1 (s)n =(ψ(s+n)−ψ(s))2 (s)n−ψ0(s+n)−ψ0(s) (s)n . 8 Here ψ(z) = Γ0(z) Γ(z) is the digamma function [5,9]. When nis a positive integer, then [2, p. 13] ψ(n) = 1 1+1 2+···+1 n−1−γ=Hn−1−γ, (13) and ψ0(n) = −1 12−1 22−···− 1 (n−1)2+ζ(2) = −Hn−1,2+ζ(2). (Hnand Hn,2are the harmonicand second order harmonic numbers, respectively. H0=H0,2= 0.) Now (11), (12) and (13) gives that ∂ ∂s2F1(1,1; s;z)s=1 =∞ X n=0 n!n!−Hn n! zn n!=−∞ X n=0 Hnzn=log(1 −z) 1−z. The last equality is straightforward (see [10]). Similarly, for the second order derivative ∂2 ∂s22F1(1,1; s;z)s=1 =∞ X n=0 n!n! H2 n n!+Hn,2 n!!zn n!=∞ X n=1 H2 nzn+∞ X n=1 Hn,2zn. (14) By Cauchy’s product, the latter sum is simply 1 1−z ∞ X n=1 zn n2=Li2(z) 1−z. The first sum can be determined easily. Note that H2 n−1=Hn−1 n2 =H2 n+1 n2−2Hn n, whence ∞ X n=1 H2 n−1zn=∞ X n=1 H2 nzn+∞ X n=1 zn n2−2∞ X n=1 Hn nzn.(15) The last sum equals to [7] ∞ X n=1 Hn nzn= Li2(z) + 1 2log2(1 −z).(16) If we temporarily introduce the function f(z) = ∞ X n=1 H2 nzn, 9